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Math Help - How to reparametrize by arc length?

  1. #1
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    How to reparametrize by arc length?

    1. Find the length of the helix f(t)= [(cost)^2), (sint)^2, 2t^2] for t E [0,10]. Reparametize this curve by arc length.
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  2. #2
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    Re: How to reparametrize by arc length?

    Quote Originally Posted by kandygirl16 View Post
    1. Find the length of the helix f(t)= [(cost)^2), (sint)^2, 2t^2] for t E [0,10]. Reparametize this curve by arc length.
    To find the length you need to integrate

    s=\int_{a}^{b}\sqrt{\left( \frac{dx}{dt}\right)^2+\left( \frac{dy}{dt}\right)^2+\left( \frac{dz}{dt}\right)^2}}}, \quad t \in [a,b]

    This gives

    s=\int_{0}^{10}\sqrt{4\sin^2(t)\cos^2(t)+4\sin^2(t  )\cos^2(t)+16t^2}dt}

    This function does not have a "nice" anti derivative.

    Did you mean the helix \mathbf{r}(t)=<\cos(t),\sin(t),2t>?
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    Re: How to reparametrize by arc length?

    Hi, thanks for the reply. No unfortunately, this is the one I meant.
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